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Translation equivalent elements in free groups. (English) Zbl 1112.20023
Two elements $$g,h$$ of a free group $$F_n$$ of rank $$n$$ are called translation equivalent if the cyclic length of $$\varphi(g)$$ equals the cyclic length of $$\varphi(h)$$ for every automorphism $$\varphi$$ of $$F_n$$. In this paper, the author proves the following Theorem: Let $$g,h\in F_n$$ be translation equivalent in $$F_n$$ and let $$w(a,b)\in F(a,b)$$ be arbitrary. Then $$w(g,h)$$ and $$w(h,g)$$ are translation equivalent in $$F_n$$. Here $$F(a,b)$$ is the free group on $$a,b$$. This theorem answers in the affirmative Problem F38b in the online version [http://www.grouptheory.info] of G. Baumslag, A. G. Myasnikov and V. Shpilrain, Open problems in combinatorial group theory. Second ed. [Combinatorial and geometric group theory. Contemp. Math. 296, 1-38 (2002; Zbl 1065.20042)]. The proof here is combinatorial.

##### MSC:
 20E05 Free nonabelian groups 20E36 Automorphisms of infinite groups 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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##### References:
 [1] DOI: 10.1023/B:GEOM.0000006579.44245.92 · Zbl 1037.57013 · doi:10.1023/B:GEOM.0000006579.44245.92
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